Absolutely! Let’s break down each part step by step to help you understand what’s going on with this cholesterol study and the related statistics questions. --- ## 1. **Understanding the Study** - **Purpose**: To see if a new diet lowers cholesterol in employees. - **Data**: Cholesterol before and after diet for 11 employees. - **Statistical Test**: Paired t-test (since measurements are from the same people before and after). **Hypotheses:** - **Null (H₀):** μd = (no difference in mean cholesterol before and after) - **Alternative (H₁):** μd ≠ (there *is* a difference) --- ## 2. **Calculating Differences** - **d = Before Diet – After Diet** (for each patient) - **d² = (d)²** You would sum up the d's and d²'s to calculate the mean and standard deviation of the differences. --- ## 3. **Decision Rule (Question 12, Image 2)** - **99% confidence** = α = .01 - **Two-tailed test** (since H₁ is μd ≠ ) - **Degrees of freedom**: n - 1 = 11 - 1 = 10 **Critical t-value for 99% confidence, df=10:** From t-tables: **±2.764** **Correct Rule:** Reject H₀ if the t-test statistic is **less than -2.764 or greater than 2.764**. So, the correct answer is: - **c) If t critical ratio is less than -2.764 t critical value reject the null hypothesis** --- ## 4. **Conclusion (Image 3)** Based on the same critical value: - If the computed t is less than -2.764, you reject the null hypothesis. The correct answer matches the critical value: - **c) The t critical ratio is less than -2.764 t critical value reject the null of no difference. T1 error is possible.** --- ## 5. **Interpretation (Image 4)** If you reject H₀, it means there **is** a significant difference in cholesterol levels. Since you expect cholesterol to go **down** after the diet, look for "decrease." - **b) There is a significant decrease in the average cholesterol differences of the eleven patients. The diets are working.** --- ## **Summary Table** | Step | Explanation | Answer (from images) | |---------------------|------------------------------------------------------------------------------------------------------|-------------------------------------| | Decision Rule | Use t-critical value for 99% confidence, two-tailed, df = 10 (±2.764) | c) from Image 2 | | Conclusion | Reject H₀ if t < -2.764; this means there is a significant difference | c) from Image 3 | | Interpretation | There is a significant **decrease** in cholesterol after the diet (diet is working) | b) from Image 4 | --- ## **Key Takeaways** - Use a **paired t-test** for before-after measurements on the same subjects. - For a **99% confidence interval** with n=11, critical t ≈ ±2.764. - If your test statistic (t) is more extreme than this value (either direction), **reject the null hypothesis**. - In context: The diet **works** if cholesterol **decreases significantly**. --- **Let me know if you want to see the calculations for the t-test or need more explanation on any step!**